finally / finite
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finally
Expresses that something happens after a long wait, a delay, or a lot of effort. It is often associated with relief.
Endlich bist du da, wir haben schon auf dich gewartet!
You're finally here; we've been waiting for you!
The word is derived from 'das Ende' (the end). This origin explains both meanings: reaching the 'Ende' (end) of a waiting period ('endlich!' — finally!) and having an 'Ende' (end), as in something being finite.
Endlich ist das Projekt fertig!
The project is finally finished!
You can recognize the meaning from its function in the sentence. As an adverb, it expresses relief ('Endlich ist Wochenende!' — It's finally the weekend!). As an adjective, it describes that something is limited ('Unsere Zeit ist endlich.' — Our time is finite.).
Endlich scheint die Sonne wieder.
The sun is finally shining again.
In the meaning 'schließlich' (finally), 'endlich' almost always conveys a strong emotion such as relief, joy, or even impatience. It emphasizes that one has waited a long time for something.
Der Zug kommt endlich, ich friere schon seit einer halben Stunde.
The train is finally coming; I've been freezing for half an hour.
The meaning 'begrenzt' (limited/finite) is used mainly in formal, philosophical, or scientific contexts. In everyday language, this usage is less common.
Die Anzahl der möglichen Lösungen ist endlich.
The number of possible solutions is finite.
finally / after all
'endlich' and 'schließlich' can both mean that something happens after a longer time or at the end of a sequence. 'endlich' puts stronger emphasis on the waiting, impatience, or relief. It often sounds personal: you had been hoping for it. 'schließlich' sounds more neutral and tends to describe the final step in a development. In addition, 'schließlich' can also introduce a reason, for example when you want to say: this should not be forgotten. In this meaning, you cannot use 'endlich'.
Nach zwei Stunden Warten kommt der Bus endlich.
After waiting for two hours, the bus finally arrives.
Wir haben lange diskutiert und schließlich eine Lösung gefunden.
We discussed it for a long time and finally found a solution.